\unsethebrew
\bchapter{מבוא}{Preface} \label{mylabel}

This is the preface.    

An integral equation of the first kind                                 
\begin{equation}                                                        
 Kf \; = \; \int_{a}^{b} k(s,t)f(t)dt \; = \; g(s)
\end{equation}       
An integral equation of the first kind                                 
\begin{equation}                                                        
K \;: \; L_2 [a,b]  \; \longrightarrow \; L_2[a,b], \;             
k \; \in L_2([a,b] \times [a,b])
\end{equation}       
is known to be an ill-posed problem.                                            
\begin{eqnarray}
R(K) \; + \; R(K)^{\perp} &=& K \;: \; L_2 [a,b] L_2([a,b] \times [a,b] \\
K \;: \; &=&L_2 [a,b]  \;     \\
Kf \;    &=& \int_{a}^{b} k(s,t)f(t)dt \; = \; g(s)
\end{eqnarray}
The following is an example.  It's but a short exerpt from 
a thesis which was subsequently submitted in a modified version.
An integral equation of the first kind                                          
\begin{equation}                                                                
 Kf \; = \; \int_{a}^{b} k(s,t)f(t)dt \; = \; g(s)
\end{equation}                
\begin{equation}                                                                
K \;: \; L_2 [a,b]  \; \longrightarrow \; L_2[a,b], \;                       
k \; \in L_2([a,b] \times [a,b])
\end{equation}                
is known to be an ill-posed problem.                                            

The solution of (1) is the least square solution of minimal norm, and           
whenever $g$ belongs to                                                         
\[ R(K) \; + \; R(K)^{\perp} K \;: \; L_2 [a,b]  \; \longrightarrow \;
L_2[a,b] k \; \in L_2([a,b] \times [a,b] \]
such
a unique solution exists. $R(K)$ is the range space of $K$ and                  
$ R(K)^{\perp} $ its orthogonal subspace.                                       
                                                                                
It is also well known that, whenever the kernel $k$ belongs to $L_2$,           
the integral operator is compact (see Groetsch \cite{1}, pp.11-12).
Thus, arbitrarily small changes in $g$ may cause arbitrarily                    
large ones                                                                      
in $f$. It has already been pointed out that this sensitivity of $f$            
will                                                                            
be the more severe, the faster the singular values of $K$ tend to zero          
(see Golberg \cite{2}, pp.36-40, Groetsch \cite{1}, p.2).
                                                                                
An integral equation of the first kind                                 
\begin{equation}                                                        
 Kf \; = \; \int_{a}^{b} k(s,t)f(t)dt \; = \; g(s)
\end{equation}       
\begin{equation}                                                        
K \;: \; L_2 [a,b]  \; \longrightarrow \; L_2[a,b], \;             
k \; \in L_2([a,b] \times [a,b])
\end{equation}       
is known to be an ill-posed problem.                                            

As an expression of this ill-posedness, for many numerical methods              
(e.g. Galerkin, collocation) the approximate solution of (1)                    
fails in general to converge to the exact solution $f$.                         
And even if it does converge,                                                   
faster                                                                          
decrease of the singular values of the operator $K$ will involve                
faster increase of the condition numbers for the matrices                       
approximating the operator. It is thus important to estimate the                
\begin{equation}                                                                
\end{equation}                
growth of these (spectral) condition numbers.                                   
                                                                                
Lower bounds of the condition numbers are given by G.M.Wing in                  
\cite{3}
for both the Galerkin and collocation methods.                                  
                                                                                
An integral equation of the first kind                                 
\begin{equation}                                                        
 Kf \; = \; \int_{a}^{b} k(s,t)f(t)dt \; = \; g(s)
\end{equation}       
\begin{equation}                                                        
K \;: \; L_2 [a,b]  \; \longrightarrow \; L_2[a,b], \;             
k \; \in L_2([a,b] \times [a,b])
\end{equation}       
is known to be an ill-posed problem.                                            

The main object of this work is bounding the condition number                   
from both sides, in terms of the singular values of the                         
operator $K$, the numerical method being a variant of                           
finite-dimensional Tikhonov regularization (see Marti                           
\cite{4}, \cite{5}, Groetsch \cite{1}).

\begin{figure}[htbp]
\epsfxsize=0.5\textwidth
\epsffile{tst1.eps}
\bcaptionff{דוגמה ראשונה לרשימת האיורים}{First English Caption to LOF}%
           {דוגמה ראשונה}{First English Caption}
\end{figure}

\begin{corolar}{$\!\!\! \bf :$}                                                 
Under the assumptions of Corollary~2 - Corollary~1                              
and Theorems~2,3 hold                                                           
when using the basis $\tilde v $ and $H_1 \; = \; H_2 \; = \;                   
L_2 [0,1] $.                                                                    
\end{corolar}                                                                   
                                                                                
\begin{figure}[htbp]
\epsfxsize=0.3\textwidth
\epsffile{tst2.eps}
\bcaptionff{דוגמא שניה לרשימת האיורים}{Second English Caption to LOF}%
           {דוגמה שניה}{Second English Caption}
\end{figure}

{\bf Proof:}                                                                    
The condition number is the same in the basis $v$ as in
$\tilde v$.
$\Box $
